Original practice · fully worked
Original variant: duration matching for a conservation reserve
A conservation trust must pay 120,000 in 11 years and discounts at 6% annually. It funds the obligation with assets having Macaulay durations 5, 10, and 18 years. The amount in the 18-year-duration asset is 1.5 times the amount in the 10-year-duration asset. Find the amount invested in the 5-year asset.
- A
15,481 - B
20,736 - C
24,512 - D
29,105 - E
36,822
Answer and calculation
Present-value and duration matching give 24,511.75 in the 5-year-duration asset, which is choice C.
Setup
Setup
Let x be the amount in the 5-year asset and y the amount in the 10-year asset. The third amount is 1.5y.
L=120000(1.06)−11=63214.50 Model
Model
Match the portfolio's dollar duration to the 11-year liability.
5x+10y+18(1.5y)=11L Compute
Compute
Eliminating x gives y = 15,481.10 and x = 24,511.75.
5(L−2.5y)+37y=11L y=15481.10 x=63214.50−2.5(15481.10)=24511.75 Answer
Answer
The 5-year-duration asset receives approximately 24,512, which is choice C.
x≈24512(C) 